Pseudoscalar heavy quarkonium production in heavy ion ultraperipheral collision

The inclusive production of pseudoscalar heavy quarkoniua (${\eta }_c,{\eta }_b$ and $B_c$) via photon-photon fusion in heavy ion ultraperipheral collision (UPC) are calculated to QCD next-to-leading order in the framework of non-relativistic QCD (NRQCD). The total cross section of ${\eta }_c$ produced in Pb-Pb UPC is 194 $\mathrm{n}\mathrm{b}$ and 1275 $\mathrm{n}\mathrm{b}$ at nucleon-nucleon c.m. energies $\sqrt{S_{\mathrm{N}\mathrm{N}}}=$ 5.52 TeV and 39.4 TeV, respectively. The cross sections for ${\eta }_b$ and $B_c$ mesons are more than two to three orders of magnitude smaller. We make a detailed phenomenological analysis on the ${\eta }_c$ production; the uncertainties caused by the renormalization scale and the charm quark mass, the cross sections in other ultraperipheral nucleon-nucleon colliding systems, and the transverse momentum distribution are discussed. At the coming HL-LHC and future FCC, the heavy ion UPC opens another door of the study on the production of heavy quarkonium.